The solution set to the inequality is x≥ -4. Therefore, we should shade everything that is greater than or equal to -4. Note that this is a non-strict inequality which means we will keep the endpoint closed.
b To solve the inequality we must first determine its boundary point(s). To do that, we will treat the inequality as if it was an equation and solve for x.
Inequality:& 2x^2-5x<12
Equality:& 2x^2-5x=12
Let's solve it by using the Quadratic Formula.
The boundaries of the inequality is x=-1.5 and =4. Let's mark these solutions on a number line. We will keep the points open since this is a strict inequality. To identify where we should shade the number line, we will also include a few test points.
By substituting x with -3, 0, and 5 in the inequality we can determine where the inequality holds true.
|c|c|c|
[-0.8em]
x & 2x^2-5x<12 & Evaluate [0.5em]
[-1em]
-3 & 2( -3)^2-5( -3)? <12 & 33 ≮ 12 * [0.5em]
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0 & 2( 0)^2-5( 0)? <12 & 0 < 12 ✓ [0.5em]
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5 & 2( 5)^2-5( 5)? <12 & 25 ≮ 12 * [0.5em]
As we can see, the inequality is true for values between the boundary points. Therefore, we should shade this region.
